Methods of seismic imaging that are known in the art use an array of seismic sources and receivers to acquire data regarding subsurface (i.e., subterranean) structures. Following a seismic stimulus (such as a detonation or mechanical shock) at a given source, each of the receivers produces a seismogram trace, i.e., a record of the seismic signal at the receiver as a function of time, due to reflections of the stimulus wave from the subsurface layers below the array. The traces from multiple receivers due to stimuli by sources at different locations are processed together in order to create an image of the layers.
As part of the imaging process, in order to increase signal/noise ratio, multiple traces from source/receiver pairs surrounding a common midpoint (CMP) are temporally aligned and then summed. (A group of traces of this sort is commonly referred to as a “gather,” and the processes of aligning and summation the traces are known as normal moveout (NMO) correction and stacking.) The alignment is meant to take into account the differences in the travel times of seismic waves between different source/receiver pairs. The dependence of travel time is commonly assumed to be hyperbolic as a function of the distance between the source and receiver, and the time shift that is applied to align the traces in order to compensate for this hyperbolic dependence is calculated by a simple formula that includes the distance between source and receiver and the velocity of waves in the media.
The assumption of hyperbolic moveout, however, is known to be an inaccurate reflection of the actual characteristics of subterranean structures, and its application leads to loss of information. Consequently, a number of alternative approaches have been proposed to provide the correct alignment between gathers of traces. For example, U.S. Pat. No. 5,103,429, whose disclosure is incorporated herein by reference, describes a method for analyzing such structures using homeomorphic imaging. (In a homeomorphic image, each element of a target is mapped one-to-one to a corresponding element of its image, so that the target object and its image are topologically equivalent.) This method is said to allow many types of stacks and corresponding images to be constructed without loss of resolution.
The Radon transform is a well-known mathematical transform that is applied in various imaging applications. Several known methods use the Radon transform for analyzing seismic data. In particular, three variants of the Radon transform, namely the slant-stack, hyperbolic and parabolic Radon transforms, have been used for seismic data analysis. The slant-stack transform is described, for example, by Treitel et al., in “Plane-Wave Decomposition of Seismograms,” Geophysics, volume 47, number 10, October, 1982, pages 1375-1401, which is incorporated herein by reference. The slant-stack transform is also addressed in U.S. Pat. No. 4,760,563, whose disclosure is incorporated herein by reference, and by Thorson and Claerbout in “Velocity-Stack and Slant-Stack Stochastic Inversion,” Geophysics, volume 50, number 12, December, 1985, pages 2727-2741, which is incorporated herein by reference.
The hyperbolic and parabolic Radon transforms are described, for example, by Foster and Mosher in “Suppression of Multiple Reflections using the Radon Transform,” Geophysics, volume 57, number 3, March, 1992, pages 386-395, which is incorporated herein by reference. The hyperbolic Radon transform is described, for example, by Mitchell and Kelamis in “Efficient Tau-P Hyperbolic Velocity Filtering,” Geophysics, volume 55, number 5, May, 1990, pages 619-625, which is incorporated herein by reference. The parabolic transform is described, for example, by Hampson in “Inverse Velocity Stacking for Multiple Elimination,” Journal of the Canadian Society of Exploration Geophysicists, volume 22, number 1, December, 1986, pages 44-55, which is incorporated herein by reference. A discrete Radon transform is described by Beylkin in “Discrete Radon Transform,” IEEE Transactions on Acoustics, Speech and Signal Processing, volume ASSP-35, number 2, February, 1987, pages 162-172, which is incorporated herein by reference.